Understanding the Relationship Between Sine Waves and Sine Angles?

[Newtons Apple]

Hello everyone. Can someone explain the relationship between the idea of a sine wave, and the idea of a sine angle? I'm getting into trig, and I hear both terms of sin tossed around, but they seem to be completely unrelated. What does the angle of the triangle have to do with a wave?

Same goes for sin, cos, tan, I know these are ratio's for angles on a right triangle, but I found these terms also applying to things unrelated to triangles

 

[tiny-tim]

Hello Newtons Apple!

Can someone explain the relationship between the idea of a sine wave, and the idea of a sine angle?

Imagine a particular spoke a wheel of radius r rotating on a fixed axle.

When it is at angle θ, trig tells you that its height (above the centre) is h = rsinθ.

If θ = ωt, with ω constant, then the height as a function of time is h = rsinωt …

this is a sine wave! ​

 

[berkeman]

Hello everyone. Can someone explain the relationship between the idea of a sine wave, and the idea of a sine angle? I'm getting into trig, and I hear both terms of sin tossed around, but they seem to be completely unrelated. What does the angle of the triangle have to do with a wave?

Same goes for sin, cos, tan, I know these are ratio's for angles on a right triangle, but I found these terms also applying to things unrelated to triangles

Hello Newtons Apple!


Imagine a particular spoke a wheel of radius r rotating on a fixed axle.

When it is at angle θ, trig tells you that its height (above the centre) is h = rsinθ.

If θ = ωt, with ω constant, then the height as a function of time is h = rsinωt …

this is a sine wave! ​

And to take it one step farther, take that spinning wheel and mark the point on the tire that lines up with the end of the spoke. Turn the wheel so you see the edge of the wheel, and you will see that marked spot going up and down (imagine you can see through the wheel to see the spot when it is on the other side). Now translate the wheel linearly to the left, and the spot will trace out a sine wave!

Click for animation -- http://www.rkm.com.au/animations/animation-sine-wave.html

 

[HallsofIvy]

I concur with tiny-tim's and berkeman's explanations of "sine wave". I have never seen the phrase "sine angle". Do you mean "sine of an angle"?

 

[Newtons Apple]

That is...sort of mind blowing..I never really though of triangles as part of a circle...and a circle part of a wave..How is this possible?? But this is only valid for triangles with angles from 0 to 90 degree's right? Also when I see sin, or cos, or tan, in an equation, they themselves don't inherently have any value right? they just denote an operation to do on another number? I was always thinking that sin, cos,tang, etc.. actually have a value associated with them..like pi.

 

[berkeman]

That is...sort of mind blowing..I never really though of triangles as part of a circle...and a circle part of a wave..How is this possible?? But this is only valid for triangles with angles from 0 to 90 degree's right? Also when I see sin, or cos, or tan, in an equation, they themselves don't inherently have any value right? they just denote an operation to do on another number? I was always thinking that sin, cos,tang, etc.. actually have a value associated with them..like pi.

Did you watch the animation? It makes it pretty clear how the sin and cos work. Your question about the tan is not so applicable to this thread/topic.

 

[DrewD]

Also when I see sin, or cos, or tan, in an equation, they themselves don't inherently have any value right? they just denote an operation to do on another number? I was always thinking that sin, cos,tang, etc.. actually have a value associated with them..like pi.

Trig functions (sinx,cosx,etc) do not have a value on their own. I see this mistake over and over again. Students treat sin as some number on its own and do algebra as though $\sin x$ means $\sin \times\, x$ and is the same as $x\times\sin$. They are functions. That is, they only have a numerical value when an argument is specified (like $x=\pi/2$).

 

[tiny-tim]

Click for animation -- http://www.rkm.com.au/animations/animation-sine-wave.html

nice!

..I never really though of triangles as part of a circle...and a circle part of a wave..How is this possible?? But this is only valid for triangles with angles from 0 to 90 degree's right?

in the red circle in berkeman's image, sinθ is the height (above the centre line) of the end of the spoke at angle θ

if θ is between 90° and 180°, that height is still positive, and shows that sin (180° - θ) = sinθ

if θ is between 180° and 360°, that height is negative, and shows that sin (360° - θ) = -sinθ